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Math Help - Possible Primes.

  1. #1
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    Lightbulb Possible Primes.

    Is it possible to either prove or disprove the following?
    If m is a positive odd integer such that 2m = 2 (mod. m(m-1)) then m is a prime.
    Examples:
    27 = 2 (mod. 42 = 7x6). 7 is prime.
    243 = 2 (mod. 1806 = 43x42). 43 is prime.
    Last edited by Stan; September 6th 2012 at 02:14 AM.
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  2. #2
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    Re: Possible Primes.

    Quote Originally Posted by Stan View Post
    Is it possible to either prove or disprove the following?
    If m is a positive odd integer such that 2m = 2 (mod. m(m-1)) then m is a prime.
    Examples:
    27 = 2 (mod. 42 = 7x6). 7 is prime.
    243 = 2 (mod. 1806 = 43x42). 43 is prime.
    Is the following proof valid?
    Theorem.
    If m is a positive odd integer such that 2m ≡ 2 (mod. m(m-1)) then m ≡ 3 (mod. 4) and m is a prime.
    Proof.
    Let 2m ≡ 2 (mod. m(m-1)) then 2m - 2 =km(m-1) for some positive integer k.
    This gives 2m-1 1 = km(m-1)/2 and since 2m-1 1 is odd, (m-1)/2 is odd implying m ≡ 3 (mod. 4).
    Let p be the least valued prime dividing (m-1)/2 . Then p is odd and 2m-1 ≡ 1 (mod. p).
    Let d be the order of 2 (mod. p).
    Then d is a positive integer < p and 2d ≡ 1 (mod. p).
    So d divides (m-1) and d divides (p-1).
    This implies there is a prime dividing d, less than p, which divides (m-1)/2 .
    This contrdicts p is the least valued prime dividing (m-1)/2 .
    Hence, m is a prime.
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  3. #3
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    Re: Possible Primes.

    The proof is invalid. However, there is an updated proof of the theorem in the post ' Primes Conjecture' which may or may not be valid.
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